WAEC SSCE - Physics - 2019

Maelezo ya Majibu

An instrument used to measure relative humidity is the hygrometer. Hygrometers measure the amount of water vapor present in the air, which is an indicator of the air's relative humidity. Hygrometers can be mechanical or electronic, and they may measure relative humidity directly or calculate it from measurements of temperature and dew point.

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We can use the formula for gravitational force between two masses: F = (G x m1 x m2)/r^2 where F = force of attraction G = gravitational constant m1 and m2 = masses of the bodies r = distance between the centers of the two bodies Plugging in the given values, we get: F = (6.67 x 10^-11 Nm^2/kg^2) x (3.0 kg) x (2.0 kg) / (0.50 m)^2 F = 1.6 x 10^-9 N Therefore, the force of attraction between the two bodies is 1.6 x 10^-9 N. Answer: 1.6 x 10^-9 N

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A ripple tank is a scientific apparatus that can be used to demonstrate the behavior of waves. It produces waves on the surface of water that can be observed and studied. I. Reflection: When waves encounter a boundary or obstacle, they can bounce back or reflect. A ripple tank can be used to demonstrate reflection by placing a barrier in the path of the waves and observing how they bounce back. II. Refraction: Waves can also change direction or bend when they pass through a medium with a different density or refractive index. A ripple tank can demonstrate refraction by placing a block in the path of the waves and observing how they change direction as they pass through it. III. Diffraction: When waves encounter an obstacle with a small opening or gap, they can bend and spread out, creating a diffraction pattern. A ripple tank can demonstrate diffraction by creating waves with a single source and observing how they spread out as they pass through a gap or opening. Therefore, the correct answer is "I, II and III".

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A capacitor is a device that is used for storing electric charges. It consists of two conductive plates separated by an insulating material called a dielectric. When a voltage is applied to the capacitor, electric charges are stored on the plates. The amount of charge that can be stored on a capacitor is proportional to the voltage applied and the capacitance of the capacitor, which is a measure of its ability to store electric charges. A transformer is a device that is used to change the voltage of an alternating current (AC) electrical supply. It works by using electromagnetic induction to transfer energy from one coil of wire to another. An ammeter is a device that is used to measure the current in an electrical circuit. It works by detecting the flow of electric charges through a conductor and measuring the resulting electric potential difference. A potentiometer is a device that is used to measure the potential difference (voltage) between two points in an electrical circuit. It works by comparing the voltage being measured to a reference voltage, and it can also be used to adjust the voltage in a circuit. Therefore, a capacitor is the device used for storing electric charges.

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The centripetal force acting on an electron moving in a circular path around the nucleus of an atom is given by the equation: F = (mv\(^2\))/r Where F is the centripetal force, m is the mass of the electron, v is the velocity of the electron, and r is the radius of the circular path. Substituting the given values into the equation, we have: F = (9.1 x 10\(^{-31}\) kg) x (2.0 x 10\(^6\) ms\(^{-1}\))\(^2\) / (6.1 x 10\(^{11}\) m) F = 6.0 x 10\(^{-8}\) N Therefore, the correct answer is: 6.0 x 10\(^{-8}\) N.

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The main idea of this question is to identify a type of wave that is not mechanical. Mechanical waves require a medium to travel through, such as a solid, liquid or gas. Sound waves and water waves are examples of mechanical waves. On the other hand, radio waves are a type of electromagnetic wave and do not require a medium to travel through. Therefore, the answer to the question is "Radio waves."

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Placing a freshly demagnetized bar magnet in the East-West direction helps ensure that the molecular magnets within the magnet remain randomly arranged. This is because, in this position, the magnet is perpendicular to the Earth's magnetic field, which would otherwise affect the alignment of the molecular magnets. By keeping the molecular magnets in a random arrangement, the magnet can be effectively demagnetized and can be magnetized again in a desired direction.

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Connecting wires, earth wires, and fuses serve as safety devices in electrical circuits, while switches are not safety devices but rather control devices that allow the flow of current to be turned on or off. Therefore, the correct answer is switch.

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Lenz's law states that the induced current in a coil always flows in a direction so as to oppose the change that causes it. For example, if there is a change in magnetic flux passing through a coil, it will induce an electromotive force (EMF) and a current will start flowing in the coil. This induced current will create a magnetic field that opposes the original change in magnetic flux. Lenz's law is a consequence of the law of conservation of energy and is a fundamental law in electromagnetism.

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The anomalous expansion of water refers to a phenomenon in which water expands instead of contracting as it cools below a certain temperature. This is a unique property of water, as most substances contract when they cool. The temperature range in which the anomalous expansion of water occurs is from 4°C to 0°C. As water cools below 4°C, it starts to contract, but as it approaches 0°C, it expands again. This means that at 0°C, water is actually less dense than it is at 4°C. Therefore, the answer is: 0°C to 4°C. This is the temperature range in which the anomalous expansion of water occurs.

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A neutral atom is an atom that has an equal number of protons and electrons. The protons are located in the nucleus of the atom, along with the neutrons, and the electrons occupy the space surrounding the nucleus. The number of protons in the nucleus of an atom determines the chemical element to which it belongs. A neutral atom has no overall charge, meaning that the positive charge of the protons is balanced by the negative charge of the electrons. Hence, the correct statement about a neutral atom is that the number of protons is equal to that of electrons.

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The vacuum in a thermoflask reduces heat loss resulting from conduction and convection by creating a space with no air molecules between the inner and outer layers of the flask, which minimizes heat transfer through these processes. Additionally, the reflective surfaces of the inner layer further minimize heat loss by reflecting any radiation back into the flask.

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The instrument that can be used to compare the magnitudes of charges on two given bodies is the Gold-leaf electroscope. When a charged object is brought close to the electroscope, the charge is transferred to the metal leaves causing them to repel each other. By comparing the extent of separation of the leaves with a known charge, the magnitude of the charge on the given bodies can be determined. Electrophorus and ebonite rods are used for charging objects, and proof planes are used to transfer charge from one body to another.

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The angle of refraction of a ray of light traveling from one medium to another can be calculated using Snell's law. Snell's law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction of the two media. Mathematically, this can be expressed as: sin θ_incidence / sin θ_refraction = n₁ / n₂ Where θ_incidence is the angle of incidence, θ_refraction is the angle of refraction, n₁ is the index of refraction of the first medium, and n₂ is the index of refraction of the second medium. Given the angle of incidence of 30° and the indices of refraction of 1.5 for the rectangular glass block and 1.0 for air, we can calculate the angle of refraction as follows: sin θ_incidence / sin θ_refraction = n₁ / n₂ sin 30° / sin θ_refraction = 1.5 / 1.0 sin θ_refraction = sin 30° / 1.5 θ_refraction = sin^-1(sin 30° / 1.5) = 19.47° Therefore, the angle of refraction of the ray of light traveling from the rectangular glass block into air is approximately 19.47°.

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The area under a velocity-time graph represents the distance traveled by an object. Velocity is the rate of change of displacement with respect to time, which means it indicates how far an object has traveled in a specific amount of time. The area under the velocity-time graph shows the product of velocity and time, which is distance. To calculate the distance traveled, we need to find the area enclosed by the velocity-time graph, which may consist of different shapes depending on the motion of the object. For example, if the object is moving at a constant velocity, the area would be a rectangle. If the object is accelerating, the area would be a triangle or a trapezoid. Therefore, the correct answer is "distance".

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The time rate of increase in velocity is called acceleration. Acceleration is a measure of how quickly an object changes its velocity, and it is calculated as the change in velocity per unit of time. Velocity is a vector quantity that describes an object's speed and direction of motion. If an object's velocity changes over time, either in magnitude (speed) or direction, it is said to be accelerating. The acceleration of an object is the rate at which its velocity changes, and it is a measure of the force acting on an object. Force is a vector quantity that describes the interaction between two objects that results in a change in motion. Force is proportional to acceleration, so a greater force will result in a greater acceleration. Momentum is a vector quantity that describes an object's mass and velocity. Momentum is conserved in an isolated system, meaning that the total momentum of the system remains constant, although the momentum of individual objects within the system may change. Speed is a scalar quantity that describes the magnitude of an object's velocity, without considering its direction. Speed is a measure of how quickly an object is moving, but it does not take into account changes in direction. Therefore, the time rate of increase in velocity is called acceleration.

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The correct statement about a moving object is "At a steady velocity, the resultant force acting on it must be equal to zero." This means that when an object is moving at a constant speed in a straight line (i.e., at a steady velocity), the net force acting on it is zero. This does not mean that there are no forces acting on the object, but rather that the forces are balanced such that they cancel each other out. If the object were to accelerate or decelerate, there would be a net force acting on it. Similarly, air resistance can be a force acting on the object but it does not need to be zero for the object to move at a steady velocity, as long as it is balanced by other forces.

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The graph shows the change in temperature of a solid as it is heated over time. At segment P, the temperature remains constant despite the application of heat. This is because the energy is being used to break the bonds between the molecules in the solid, rather than increasing their kinetic energy and temperature. This process is called melting, which is the transition from solid to liquid phase. At segment Q, the temperature again remains constant, but this time at a higher level than at segment P. This is because the energy is being used to break the bonds between the molecules in the liquid, rather than increasing their kinetic energy and temperature. This process is called boiling, which is the transition from liquid to gas phase. Therefore, the correct answer is "Melting and boiling".

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In an elastic collision, both momentum and kinetic energy are conserved. Momentum is defined as the product of an object's mass and velocity, and it is a measure of an object's motion. In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. This means that the momentum of the objects involved in the collision is conserved. Kinetic energy is defined as the energy an object possesses due to its motion. In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. This means that the kinetic energy of the objects involved in the collision is conserved. In contrast, potential energy is a measure of an object's potential to do work, and it is related to an object's position or configuration. In an elastic collision, potential energy is not conserved because the objects change position or configuration during the collision. Therefore, in an elastic collision, momentum and kinetic energy are conserved, while potential energy is not conserved.

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The formula for the speed of a wave is given by: v = λf where v is the speed of the wave, λ is the wavelength, and f is the frequency. We are given the frequency of the pulse as 9 Hz and the time it takes to travel a distance of 4.5 m as 0.6 s. We can use these values to calculate the speed of the pulse as follows: v = d/t = 4.5 m/0.6 s = 7.5 m/s Now, we can use the formula v = λf to calculate the wavelength of the pulse: λ = v/f = 7.5 m/s / 9 Hz = 0.83 m Therefore, the wavelength of the pulse is 0.83 m. Answer option (C) is correct.

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Out of the given options, the quantity that is a vector is momentum. A vector quantity has both magnitude and direction. Momentum is the product of mass and velocity, and it has both magnitude and direction. On the other hand, volume, energy, and speed are scalar quantities that have only magnitude but no direction associated with them.

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The property that is not exhibited by sound waves is polarization. Polarization is a property of transverse waves, which are waves that vibrate perpendicular to the direction of their motion. Sound waves, on the other hand, are longitudinal waves, which means that they vibrate parallel to the direction of their motion. Since sound waves are not transverse, they cannot exhibit polarization. Diffraction, interference, and reflection are all properties of sound waves. Diffraction is the bending of sound waves around obstacles or through openings. Interference occurs when two or more sound waves interact with each other, either constructively or destructively. Reflection is the bouncing back of sound waves when they hit a surface. In summary, sound waves are longitudinal waves and cannot exhibit polarization, but they can exhibit other properties such as diffraction, interference, and reflection.

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The heat needed to change the temperature of 60g of ice at 0°C to 80°C can be calculated using the equation for heat transfer: Q = m * c * ΔT Where Q is the heat transferred, m is the mass of the substance, c is the specific heat capacity of the substance, and ΔT is the change in temperature. First, the heat needed to melt the ice can be calculated using the specific latent heat of fusion: Q_fusion = m * L_f Where L_f is the specific latent heat of fusion. Q_fusion = 60g * 3.36 x 10^5 J/kg = 2.016 x 10^5 J Next, the heat needed to raise the temperature of the water from 0°C to 80°C can be calculated using the specific heat capacity of water: Q_heat = m * c * ΔT Q_heat = 60g * 4.2 x 10^3 J/kg/K * 80°C = 2.016 x 10^5 J The total heat needed to change the temperature of the ice to 80°C is the sum of the heat needed to melt the ice and the heat needed to raise the temperature of the water: Q = Q_fusion + Q_heat = 2.016 x 10^5 J + 2.016 x 10^5 J = 4.032 x 10^5 J = 40.32 kJ Therefore, the quantity of heat needed to change the temperature of 60g of ice at 0°C to 80°C is 40.32 kJ.

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The angular speed of the electron can be determined using the formula: ω = v / r where ω is the angular speed, v is the velocity of the electron, and r is the radius of the circular path. Substituting the given values, we get: ω = (2.0 x 10^6 m/s) / (6.1 x 10^11 m) ω = 3.28 x 10^(-5) rad/s Therefore, the angular speed of the electron is 3.28 x 10^(-5) rad/s. Hence, the correct option is: 3.28 x 10^16 rad s^(-1)

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The change in the area of a rectangular piece of iron due to an increase in temperature can be calculated using the formula for linear expansion: ΔL = α * L * ΔT Where ΔL is the change in length, α is the linear expansivity, L is the original length, and ΔT is the change in temperature. The new length of the iron can be calculated by adding the change in length to the original length: L' = L + ΔL The new area of the iron can be calculated by using the new length and width: A' = L' * W First, the change in length of the iron can be calculated for both the length and the width: ΔL = α * L * ΔT = 1.2 x 10^-5 K^-1 * 4 cm * 100°C = 4.8 x 10^-3 cm L' = L + ΔL = 4 cm + 4.8 x 10^-3 cm = 4.0048 cm ΔL = α * L * ΔT = 1.2 x 10^-5 K^-1 * 3 cm * 100°C = 3.6 x 10^-3 cm W' = W + ΔL = 3 cm + 3.6 x 10^-3 cm = 3.0036 cm The new area of the iron can be calculated using the new length and width: A' = L' * W' = 4.0048 cm * 3.0036 cm = 12.0288 cm^2 Therefore, the new area of the metal after a temperature increase of 100°C is 12.0288 cm^2.

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The nucleon number of an atom is the total number of protons and neutrons in its nucleus, while the proton number is the number of protons. Therefore, to determine the number of neutrons in an atom, we can subtract the proton number from the nucleon number. In this case, the nucleon number is 238 and the proton number is 92. Number of neutrons = Nucleon number - Proton number Number of neutrons = 238 - 92 Number of neutrons = 146 Therefore, the answer is: 146 neutrons.

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The resistance (R) of a wire is proportional to the temperature (T) in degrees Celsius. The equation of the resistance-temperature relationship is given by: R = R0 (1 + αT) Where R0 is the resistance at 0°C, α is the temperature coefficient of resistance and T is the temperature in degrees Celsius. To solve this problem, we can use the resistance-temperature relationship to find α, and then use it to find the temperature at the given resistance. First, we need to find the value of α. We can use the two resistance values given to set up two equations: R1 = R0 (1 + αT1) R2 = R0 (1 + αT2) where R1 and R2 are the resistances at temperatures T1 and T2, respectively. Substituting the given values: 3.5 = R0 (1 + α(0)) 10.52 = R0 (1 + α(100)) Solving for R0: 3.5 = R0 1 + α(100) = 10.52/R0 Rearranging the second equation: α = (10.52/R0 - 1)/100 Substituting R0 = 3.5: α = (10.52/3.5 - 1)/100 = 0.0032/°C Now we can use the resistance-temperature relationship to find the temperature at the given resistance: R = R0 (1 + αT) 7.5 = 3.5 (1 + 0.0032T) Solving for T: T = (7.5/3.5 - 1)/0.0032 = 57°C Therefore, the temperature of the wire when its resistance is 7.5Ω is 57°C. The correct option is 57 \(^o\)C.

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(a) (i) Force is a push or pull which changes, or tends to change, the state of rest or uniform motion of a body in a straight line.

The S.I. unit of force is the newton (N).

(ii) The two types of solid friction are:

1. Static friction;
2. Dynamic (sliding or kinetic) friction.

(b) (i) Velocity-time graph

The successive points plotted are  t time  tand  tvelocity  tas follows: \[(0,30),\ (20,30),\ (26,18),\ (36,18),\ (41,0).\]

(ii) Deceleration during the last 5 s

For the final stage,

\[ a=\frac{v-u}{t}=\frac{0-18}{5}=-3.6\ \text{m s}^{-2}. \]

Therefore, the deceleration is \(3.6\ \text{m s}^{-2}\).

(iii) Total distance covered

The total distance is the area under the velocity-time graph.

\[ \text{Total distance}=600+144+180+45=\boxed{969\ \text{m}}. \]

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(a)(i) Parts of the photocell (standard evacuated photocell): A is the photocathode (emitter) - a curved metal plate coated with a photosensitive material. B is the anode (collector) - a small metal rod or ring placed in front of the cathode.

(a)(ii) Functions: A (photocathode) emits electrons when light of sufficient frequency falls on it. B (anode) collects the emitted photoelectrons, so completing the circuit and allowing a photocurrent to flow.

(a)(iii) Terms in Einstein's equation \(E = hf - W_0\): \(E\) is the maximum kinetic energy of the emitted photoelectron; \(hf\) is the energy of the incident photon (h = Planck's constant, f = frequency); \(W_0\) is the work function, the minimum energy needed to release an electron from the metal surface.

(b) Frequency of the photon: Photon energy \(= KE_{max} + W_0 = 1.97 + 1.32 = 3.29\,\text{eV}\).

\[ E = 3.29\times1.6\times10^{-19} = 5.264\times10^{-19}\,\text{J}. \]

\[ f = \frac{E}{h} = \frac{5.264\times10^{-19}}{6.6\times10^{-34}} \approx 7.98\times10^{14}\,\text{Hz}. \]

(c)(i) Half-life: The half-life of a radioactive element is the time taken for half the atoms (nuclei) originally present in a sample to decay.

(c)(ii) Graph of \(N = N_0 e^{-\lambda t}\): Plot \(N\) (number of undecayed nuclei) on the vertical axis against time \(t\). The curve starts at \(N_0\) and falls exponentially towards zero. The half-life \(t_{1/2}\) is read on the time axis at the point where \(N = N_0/2\).

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(a) Free electrons: These are the outermost (valence) electrons of the atoms of a conductor or semiconductor that are so loosely bound that they become detached from their parent atoms and are able to move about randomly throughout the material. Because they are mobile and carry negative charge, they act as the charge carriers responsible for the flow of electric current when a potential difference is applied.

(b) Holes: A hole is the vacancy (empty space) left behind in the covalent bond structure of a semiconductor when an electron leaves its position. It behaves as a mobile carrier of positive charge, equal in magnitude to the charge on an electron. When a neighbouring electron moves in to fill the vacancy, the hole appears to move in the opposite direction, so holes constitute a flow of positive charge and contribute to conduction (dominant in p-type material).

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(a) Diffusion: Diffusion is the net movement of molecules of a substance from a region of higher concentration to a region of lower concentration, as a result of the random motion of the molecules, until the molecules are evenly distributed.

(b) Two factors that affect the rate of diffusion:

1. Temperature - a higher temperature gives the molecules greater average kinetic energy, so they move faster and diffuse more rapidly.
2. Relative molecular mass (density) of the substance - lighter (less dense) molecules move faster and therefore diffuse more quickly than heavier ones.

(The concentration gradient and the medium of diffusion, gas versus liquid, are also acceptable factors.)

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The stretched rubber stores elastic potential energy, which is converted into the kinetic energy of the stone when it is released.

Data: mass \(m = 20\ \text{g} = 0.02\ \text{kg}\); extension \(x = 5\ \text{cm} = 0.05\ \text{m}\); force constant \(k = 200\ \text{Nm}^{-1}\).

Elastic potential energy stored in the rubber:

By conservation of energy, this equals the kinetic energy of the stone as it leaves the catapult:

The stone leaves the catapult with a speed of \(5\ \text{ms}^{-1}\).

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The three magnetic elements that fully determine the earth's magnetic field at a point are:

1. The angle of declination - the angle between the geographic (true) meridian and the magnetic meridian at the point.
2. The angle of dip (inclination) - the angle that the total earth's magnetic field makes with the horizontal at the point.
3. The horizontal component of the earth's magnetic field intensity \((B_H)\) - the strength of the horizontal part of the field at the point.

Together these three quantities specify both the direction and the magnitude of the earth's field at any location.

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(a)(i) Atomic spectra are the characteristic discrete lines of definite wavelengths or frequencies emitted or absorbed by atoms when electrons move between quantised energy levels.

(a)(ii) An emission spectrum consists of bright lines on a dark background. It is produced when excited electrons fall from higher to lower energy levels, emitting photons. An absorption spectrum consists of dark lines on a bright continuous background. It is produced when atoms absorb photons of particular energies, causing electrons to move from lower to higher energy levels.

(b) Transition from \(n=3\) to \(n=1\)

From the energy-level diagram, \(E_3=-1.51\text{ eV}\) and \(E_1=-13.6\text{ eV}\).

(i) Energy of the photon

\[\Delta E=E_3-E_1=-1.51-(-13.6)=12.09\text{ eV}.\]

\[\Delta E=12.09\times1.6\times10^{-19}=1.934\times10^{-18}\text{ J}.\]

Energy of the photon = \(1.93\times10^{-18}\text{ J}\) \((=12.09\text{ eV})\).

(ii) Frequency of the photon

Using \(E=hf\),

\[f=\frac{E}{h}=\frac{1.934\times10^{-18}}{6.6\times10^{-34}}=2.93\times10^{15}\text{ Hz}.\]

Frequency = \(2.93\times10^{15}\text{ Hz}\).

(iii) Wavelength of the photon

\[\lambda=\frac{c}{f}=\frac{3.0\times10^8}{2.93\times10^{15}}=1.02\times10^{-7}\text{ m}.\]

Wavelength = \(1.02\times10^{-7}\text{ m}\) (about \(102\text{ nm}\)).

(c)(i) Soft X-rays have relatively longer wavelengths, lower frequencies and energies, and low penetrating power. Hard X-rays have shorter wavelengths, higher frequencies and energies, and greater penetrating power.

(c)(ii) Circuit symbol for a p-n junction diode

(c)(iii) A semiconductor is doped to increase the number of charge carriers, thereby increasing its electrical conductivity (or reducing its resistivity).

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Observable phenomena in which electromagnetic waves behave like particles
-  photoelectric effect;
-  Compton effect (Compton scattering);
-  pair production;
-  blackbody radiation.

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(a) Identifying the labelled quantities (standard projectile diagram):

• P is the velocity of projection (the initial speed with which the body is launched).
• β is the angle of projection (the angle the initial velocity makes with the horizontal).
• H is the maximum height reached by the projectile.
• R is the horizontal range (the horizontal distance travelled before the projectile returns to the launch level).

(b) Relationship between P, g and R_max:

The range of a projectile launched at speed \(P\) and angle \(\beta\) is

\[ R = \frac{P^{2}\sin 2\beta}{g}. \]

The range is greatest when \(\sin 2\beta = 1\), i.e. when \(\beta = 45^{\circ}\). Then

\[ R_{max} = \frac{P^{2}}{g}. \]

Equivalently, \(P = \sqrt{g\,R_{max}}\).

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(a)(i) Latent heat: Latent heat is the quantity of heat energy absorbed or given out by a substance during a change of state (for example melting or boiling) at constant temperature. The heat is used to change the arrangement/spacing of the molecules rather than to raise the temperature.

(a)(ii) Two factors affecting the rate of evaporation: (1) Temperature of the liquid; (2) Surface area of the liquid exposed. (Draught/wind speed and humidity of the surrounding air are also acceptable.)

(b)(i) Clay pot cooler than closed plastic container: A clay pot is porous, so water seeps to the outside and evaporates. Evaporation takes latent heat of vaporisation from the remaining water, cooling it. In the sealed plastic container no evaporation (or escape of vapour) can occur, so no cooling takes place and the water stays warmer.

(b)(ii) Faster cooking in a pressure cooker: In a sealed pressure cooker the steam produced raises the pressure above the water. Increased pressure raises the boiling point of the water above 100 °C, so the food is cooked at a higher temperature, which speeds up cooking.

(c) Two effects of heat on a substance: (1) It causes expansion (increase in size); (2) It causes a rise in temperature or a change of state. (Change in electrical resistance or chemical change are also acceptable.)

(d) Power of heater \(P = VI = 40\times12 = 480\,\text{W}\).

(i) Latent heat of fusion: Heat supplied to melt the body \(= P t_1 = 480\times1400 = 672000\,\text{J}\). This equals \(mL\):

\[ L = \frac{Pt_1}{m} = \frac{672000}{1.5} = 4.48\times10^{5}\,\text{J kg}^{-1}. \]

(ii) Specific heat capacity: Heat supplied in the extra 72 s \(= 480\times72 = 34560\,\text{J} = mc\,\Delta\theta\):

\[ c = \frac{34560}{1.5\times60} = 384\,\text{J kg}^{-1}\,\text{K}^{-1}. \]

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(a) Principle of operation of fibre optics: An optical fibre works on the principle of total internal reflection. It consists of a transparent core of high refractive index surrounded by a cladding of lower refractive index. Light entering one end strikes the core-cladding boundary at an angle greater than the critical angle, so it is totally internally reflected. The light is repeatedly reflected along the fibre and is thereby guided from one end to the other with very little loss, even when the fibre is bent.

(b) Two applications of fibre optics in medicine:

1. Endoscopy - fibre-optic endoscopes are used to view and examine internal organs and cavities (for example the stomach, intestines and interior of the lungs) without major surgery.
2. Keyhole (minimally invasive) surgery and laser delivery - fibres are used to transmit laser light to internal tissues for cutting, cauterising or treating them, and to illuminate the operating site during such procedures.

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(a)(i) Newton's Law of Universal Gravitation: Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

\[ F = \frac{G m_1 m_2}{r^{2}} \]

(a)(ii) Gravitational field: A gravitational field is a region of space in which a body of mass experiences a force of attraction. Its strength at a point is the force per unit mass acting on a small mass placed at that point.

(b)(i) Relationship between G and g: Consider a body of mass \(m\) resting on the earth's surface. The earth (mass \(M\), radius \(R\)) attracts it with a force which, by Newton's law, is

\[ F = \frac{G M m}{R^{2}}. \]

This same force is the weight of the body, \(F = mg\). Equating the two:

\[ mg = \frac{G M m}{R^{2}} \quad\Rightarrow\quad g = \frac{G M}{R^{2}}. \]

(b)(ii) Assumptions: (1) The earth is a perfect sphere of uniform density, so its whole mass may be taken to act at its centre. (2) The body is small compared with the earth, and effects such as the earth's rotation and air resistance are neglected.

(c) Force between the star and the earth:

\[ r = 1.50\times10^{8}\,\text{km} = 1.50\times10^{11}\,\text{m} \]

\[ F = \frac{G M_{star} M_{earth}}{r^{2}} = \frac{(6.67\times10^{-11})(2.00\times10^{30})(5.98\times10^{24})}{(1.50\times10^{11})^{2}} \]

\[ F = \frac{7.98\times10^{44}}{2.25\times10^{22}} \approx 3.55\times10^{22}\,\text{N}. \]

(d)(i) Escape velocity: The minimum velocity with which a body must be projected from the surface of the earth (or a planet) so that it completely overcomes the gravitational pull and escapes without ever returning. \(v_e = \sqrt{2gR}\).

(d)(ii) Two differences between g and G:

• g (acceleration of free fall) varies from place to place on the earth and from planet to planet, whereas G (universal gravitational constant) has the same value everywhere in the universe.
• g is a vector with unit m s^-2, while G is a scalar constant with unit N m² kg^-2.